A Method for Evaluating the Formability of TailorRolled Blank (TRB) by Means of the Forming LimitMargin Field Graph in Forming ProcessDa Cai 

Hunan UniversityHang Ou 

Hunan UniversityMing Hu 

Hunan UniversityGuangyao Li 

Hunan UniversityJunjia Cui  ( cuijunjia@hnu.edu.cn )

Hunan University https://orcid.org/0000-0001-8004-6030

Research Article

Keywords: Tailor rolled blank (TRB), Forming limit margin �led graph, Quantitative evaluation offormability, Cracking prediction

Posted Date: April 28th, 2021

DOI: https://doi.org/10.21203/rs.3.rs-447084/v1

License: This work is licensed under a Creative Commons Attribution 4.0 International License.  Read Full License

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A method for evaluating the formability of tailor rolled blank

(TRB) by means of the forming limit margin field graph in forming

process

Da Cai, Hang Ou, Ming Hu, Guangyao Li, Junjia Cui *

State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body,

Hunan University, Changsha, 410082, China

Abstract

Tailor rolled blank (TRB) with graded thickness has shown great potential in the

automobile field. Using traditional forming limit diagrams (FLDs) to evaluate TRB

formability is challenging due to thickness variations. In this paper, a 3D forming limit

surface (FLS) considering the influence of thickness was obtained. A numerical model

was developed to predict final strains. Moreover, a forming margin was denoted and

calculated to generate the forming limit margin field graph for quantitative evaluation

of the TRB formability. Results showed that as the punch travel increased, the forming

margin value decreased. As the travel changed from 35.2 mm to 37.4 mm, the

corresponding forming margin value changed from 0.002 to -0.024. The formability

declined, and the specimen eventually cracked on the thinner side. Besides, the

deformation and strain paths predicted by simulation agreed well with those measured

from formed part, which indicated that the final strains used in formability evaluation

were reliable. The method was suitable for quantitative evaluation of the formability

and predicting the cracking position in TRB forming.

Keywords: Tailor rolled blank (TRB); Forming limit margin filed graph; Quantitative

evaluation of formability; Cracking prediction

* Corresponding author: Tel.: +86 731 88664001; Fax: +86 731 88822051; Email: cuijunjia@hnu.edu.cn

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1. Introduction

With the increasing demand for high-performance materials in the automobile and

aerospace industries, the application of lightweight materials such as aluminum alloys

and high-strength steels has become more and more extensive. Advanced

manufacturing technologies open up new possibilities for the application of lightweight

materials [1-3]. Tailor rolled blank (TRB) with graded thickness obtained by rolling

was regard as a potential lightweight metal sheet [4]. As shown in Fig. 1, the TRB

rolling process was continuous. The constant thickness zone and thickness transition

zone were made by adjusting the roll gap online according to actual needs. There was

no abrupt change in thickness along the rolling direction, so that TRB could have good

surface quality and excellent formability [5-7]. Due to the remarkable advantages of

TRB, some well-known automotive manufacturers have introduced TRB in their

products to reduce the vehicle weight [8-10].

Fig. 1 A schematic of TRB technology (adapted from Ref. [4])

The prediction of material formability has been a key requirement in the forming

process of TRB applications. The formability prediction aimed to increase productivity

by reducing the number of failures. To predict the cracking of TRB in forming process,

criteria for the evaluation of TRB formability should be established. One way to

evaluate the formability of a formed part was by thinning. In industrial applications, the

formed part with a thinning rate of more than 15% was generally considered scrap.

However, the evaluation only through thinning was not sufficient, it was also inaccurate

3

without considering the sheet material parameters. The forming limit diagram (FLD)

worked as another effective limiting criterion to describe the deformation degree of

materials without cracking [11, 12]. It represented the largest major and minor strains

that a blank could withstand before necking or cracking occurred in all possible

combinations of strain paths during forming. Theoretically, the FLD used in TRB

forming should contain an infinite number of forming limit curves (FLCs).

To obtain the FLCs, empirical methods and experiments could be used. Ghazanfari

et al. [13] proposed an empirical law in terms of sheet thickness to determine FLD

without experimental data. Abspoel et al. [14] derived predictive equations of the FLCs

from the statistical relations between the measured FLC points and the mechanical

properties. Kim et al. [15] used various constitutive models to predict the formability

of high strength steels. Although the cost of experiments was higher than that of

empirical methods, the experimental results were more realistic and direct. The

Nakazima test was a standard experiment that provided the sheet formability

information. Holmberg et al. [16] developed a test method which can quickly determine

the forming limit in plane. The tests were carried out in a tensile testing machine.

Previous studies on the formability of TRB were mainly focused on the forming

limit in deep drawing process. Mayer et al. [8] used TRB to increase the maximum deep

drawing depth. They evaluated the results through two-dimensional FLD of the sheet

with constant thickness. Zhang et al. [9] discussed forming limit of TRB square box

during the drawing process. The formability evaluation criteria they used were

maximum drawing depth and thinning. These studies did not take into account the

continuous thickness variation in the thickness transition zone. Moreover, there was

little quantitative evaluation of formability in these studies. The thickness of TRB was

continuously changing. The mechanical properties and formability of different

thickness were different [17]. Quantitative assessment of the safety degree and the

cracking degree can contribute to the design and production of TRB applications.

Therefore, it is necessary to establish a criterion by considering the thickness effect in

the TRB forming process. The criterion should have the ability to evaluate the

formability quantitatively.

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In this paper, the forming limit margin field graph was proposed to quantify the

formability and predict cracking. The method could be divided into four steps. Firstly,

a 3D forming limit surface (FLS) was established based on the FLCs in the constant

thicknesses zone. Secondly, uniaxial tensile tests and the Lagrange interpolation

method were used to obtain the constitutive relation of the TRB. Thirdly, a numerical

model was established to obtain major and minor strains of all elements. Finally,

forming margins were calculated based on major strains, minor strains and the FLS.

The forming limit margin field graph was established and verified by the experiment.

2. Forming limit margin field graph

2.1. 3D forming limit surface

The FLD obtained by experiments or empirical methods has proven to be a good

failure criterion. In order to better analyze the deformation process of TRB, the

traditional 2D FLCs were replaced by the 3D FLS. The process of obtaining the FLS

was divided into two steps. Firstly, the FLCs in the constant thicknesses zone were

obtained by the forming limit test. Secondly, upon the basis of the FLCs, the FLS was

constructed by polynomial fitting.

The FLCs of HC340LA under thicknesses of 1.2 mm, 1.4 mm, 1.6 mm and 1.8

mm were selected [17]. Nine types of geometrics in each group were applied to obtain

different strain paths and the FLC. The forming limit tests combined with the 3D Digital

Image Correlation (DIC) technology were carried out. The limit strains were

determined by the combination of position dependent method and DIC technology. The

FLCs were obtained by fitting.

To obtain the forming limit in the thickness transition zone, the polynomial

interpolation method was adopted. The 3D FLS of TRB was obtained by fitting the four

FLCs. In the method, the minor strain was defined as x, the thickness was defined as y

and the major strain was defined as z. The relationship between the fitting result and

the actual value can be defined by the following equation:

, ,fittingz x y z x y (1)

where zfitting (x, y) represents fitted polynomial, and ε represents synthetic error. For

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polynomial fitting surface, the goodness of fit was used to evaluate the surface fitting

accuracy. The coefficient of determination (R-square) and the root mean squared error

relative error (RMSE) were important statistics to measure goodness of fit.

R-square can be defined as

2

1

2

1

1 1, 2, ...,

n

i

ien

i i

i

R square i n

z z

(2)

RMSE can be defined as

2

1

1 1, 2, ...,

n

i e

i

RMSE i nn

(3)

where εi is the synthetic error value at the test point, iz is the mean of the measured

major strain data. The polynomial used to fit the 3D FLS can be defined by Eq. (4):

2 3 2

00 10 01 20 11 30 21,fittingz x y p p x p y p x p xy p x p x y (4)

where p00, p10, p01, p20, p11, p30 and p21 are the polynomial fitting coefficients.

The fitted 3D FLS is shown in Fig. 2. The polynomial fitting coefficients of the

3D FLS are listed in Table 1. R-square = 0.998 and RMSE = 0.002564 indicate that the

fitting result has a good accuracy. Thus, the 3D FLS could be used to determine the

forming limit of any specific thickness in the TRB.

Fig. 2 3D FLS obtained by polynomial fitting

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Table 1 Polynomial fitting coefficients of the 3D FLS

Coefficient Value Coefficient Value

p00 0.05833 p21 -0.3296

p10 -0.08886 p30 -4.458

p01 0.1174 R-square 0.998

p20 4.009 RMSE 0.002564

p11 0.07294

2.2. Forming margin field graph

Cracking of the formed part could be judged qualitatively by the traditional FLD,

while the safety degree of the non-cracking area and the cracking degree of the crack

area could not be evaluated quantitatively. In previous studies, Naceur et al. [18]

defined a constant as the “safety margin”, which could be expressed as the difference

between the major strain of the formed element and the corresponding major strain on

the “secure FLC”. Wei et al. [19] proposed an efficient method for controlling the

forming quality, in which the distance between the strain state point and the FLC was

regard as one of the constraints in the process of the blank metal forming optimization.

In order to evaluate the formability of blank quantitatively, the forming margin was

proposed in this paper. The forming margin refers to the formability of the blank after

a certain degree of plastic deformation.

The minimum distance between the final strain state point of the element and the

FLS at the same initial special thickness was defined as absolute value of the forming

margin. In detail, the strain state and initial thickness could be represented as P0 (x0, y0,

z0) in the coordinate space. The strain state and initial thickness on the FLS could be

represented as P (x, y, z), where, x, y and z represent the minor strain, the thickness and

the major strain respectively.

The thickness y0 was brought into the fitting equation of the FLS to obtain the FLC

corresponding to the thickness y0. Then the FLS could be simplified as:

0,fittingz x y z (5)

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P (x, y0, z) was a point on the FLC. The distance between the point P0 and the point

P was the minimum distance between the point P0 and the FLC. The point P0 (x0, y0, z0)

and the point P (x, y0, z) satisfied the following conditions with on the x-z plane:

0-1

p ppk k (6)

where kp is the tangent slope of the FLC at the point P; kPP0 is the slope of the line PP0.

Eq. (6) can be equivalent to the following equation:

0

0 0

,- -

fittingd z x y

z z x xdx

(7)

By combining Eqs. (5) and (7), the point P (x, y0, z) could be obtained. Then the

shortest distance between the point P0 and the FLC could be calculated by the following

Eq. (8), which was expressed as d.

2 2

0 0- -d z z x x (8)

The forming margin could be expressed as:

0 0 0

0 0 0

0 0 0

, - 0

0 , - 0

, - 0

fitting

fitting

fitting

d z x y z

M z x y z

d z x y z

(9)

zfitting(x0, y0) - z0 > 0 indicates that the strain state point is below the FLS. As shown

in Fig. 3, P01 is below the FLS, P1 is on the FLS. The distance of line P1P01 is the

minimum distance between P01 and the FLC corresponding to the thickness y01. The

distance of line P1P01 is defined as the forming margin at the thickness y01. As the d

decreases, the forming margin decreases and the formability of the blank decreases.

zfitting(x0, y0) - z0 < 0 indicates that the strain state point is above the FLS. As shown

in Fig. 3, P02 is above the FLS; P2 is on the FLS. The distance of line P2P02 is the

minimum distance between P02 and the FLC corresponding to the thickness y02. The

opposite value of the distance is defined as the forming margin at thickness y02, which

represents that cracking occurs during the forming process at this position. As the d

increases, the -d decreases, which represents the forming margin decreases and the

cracking degree of the blank increases.

zfitting(x0, y0) - z0 = 0 indicates that the strain state point is on the FLS, the blank is

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in a critical state of cracking.

Fig. 3 3D FLD and definition of forming margin

The forming margin values of all elements in the simulation model were calculated.

Then the color gradient was used to characterize the difference in forming margin of

each element on the part. The forming margin filed graph was generated. The

calculation process for all elements on the part is shown in Fig. 4.

Fig. 4 Flow chart of forming margin calculation for all elements

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In the forming margin field graph, the different margins of elements on the part

were expressed in different colors. The same color gradient was applied to the strain

state points of the corresponding elements in the 3D FLD. The 3D FLD was used

together with the forming margin field graph for TRB forming. The combination of two

was called the forming limit margin field graph. By the forming limit margin field graph,

the formability of the TRB could be quantified and the cracking in the forming process

could be predicted.

3. Simulation modeling and experiment

3.1. Materials

The investigated TRB was made of the HC340LA cold rolled steel. Some

mechanical properties [17] were listed in Table 2. The thickness of the investigated TRB

was 1.2/1.6 mm.

Table 2 Some mechanical properties of tested TRB

Young’s modulus (GPa) Poisson’s ratio Density (kg/m3)

210 0.3 7800

As a simple and practical mechanical properties test method, uniaxial tensile test

has been widely used. However, different strain hardening of the TRB could lead to

different mechanical properties during the rolling process. Different mechanical

properties should be considered in the different zones [20]. One way to obtain the

mechanical properties in the thickness transition zone was by using the interpolation

method based on the uniaxial tensile data in the constant thickness zone [9, 21].

The mechanical properties in the constant thickness zone were obtained by

uniaxial tensile tests [17]. The true stress-true strain curves in the constant thickness

zone were shown in Fig. 5(a). It was seen that the mechanical properties of two

thicknesses had difference. The flow stress of the blank with the thickness of 1.2 mm

was higher than the flow stress of the blank with the thickness of 1.6 mm. To obtain the

mechanical properties in the thickness transition zone, the Lagrange polynomial

interpolation method was adopted based upon the mechanical properties in the constant

thickness zone. Lagrange interpolation polynomial can be expressed as [22]:

10

0

x x xn

n i i

i

p f l

(10)

where li(x) is Lagrange primary function, it can be expressed by following equation:

0 1 1 1

0 1 1 1

... ... 0, 1, ...,

... ...

i i n

i

i i i i i i i n

x x x x x x x x x xl x i n

x x x x x x x x x x

(11)

li(x) is an n-degree polynomial and has the following rule:

1

, 0,1,..., .0

i ij

i jl x i j n

i j

(12)

l0(x), l1(x), …, ln(x) are the Lagrange primary functions of x0, x1, …, xn. They are

linearly independent quantities.

The 1.2/1.6 mm thickness transition zone was divided into 10 parts. Based on

uniaxial tensile testing data in the 1.2 mm zone and 1.6 mm zone, mechanical properties

of 1.22 mm, 1.26 mm, 1.30 mm, 1.34 mm, 1.38 mm, 1.42 mm, 1.46 mm, 1.50 mm, 1.54

mm and 1.58 mm were obtained using Lagrange polynomial interpolation method. The

mechanical properties of these interpolation points were combined to characterize the

mechanical properties in the thickness transition zone, as shown in Fig. 5(b). Thus, the

constitutive relation of TRB was completed by means of the combination of uniaxial

tensile tests and Lagrange polynomial interpolation method.

Fig. 5 True stress-true plastic strain curves of the TRB: (a) in the constant thickness

zone; (b) in the thickness transition zone

3.2. Numerical modeling

The TRB bulging model was developed by using the HyperMesh, Dynaform and

Matlab software. The schematic of TRB bulging modeling is shown in Fig. 6. The

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blank was modeled using four-node quadrilateral Belytschko-Tsay shell elements,

which had 5 integration points in the thickness direction. In the model, the keyword

*ELEMENT_SHELL_THICKNESS was used to change the thickness of the shell

elements at the four nodes. As shown in Fig. 8, T1 - T4 represented the thicknesses at

nodes N1 - N4, respectively. For elements in the constant thickness zone, the

thicknesses were same at the four nodes (T1 = T2 = T3 = T4). For elements in the

thickness transition zone, the interpolation method was employed to obtain the

thicknesses according to the coordinate of the nodes (T1 = T2, T3 = T4). Each part was

assigned the corresponding mechanical properties (see Fig. 5).

Fig. 6 Schematic of bulging modeling

In an ideal model, the thickness transition zone should be discretized into countless

parts, the element size would be small. However, the small element size would result

in high computational cost and numerical instability [21]. It was found that an element

size of 4 mm × 4 mm was sufficient in the thickness transition zone, while the thickness

transition zone was discretized into 10 parts. Within the constant thickness zone, the

1.6 mm zone and the 1.2 mm zone were set to the other two parts, respectively. The

average size of an element was 4 mm × 3.89 mm. The total number of the blank

elements was 2070. The die, punch and holder were defined as rigid bodies, which were

meshed automatically in Dynaform software. The total number of elements for the

entire model was 19235. The major contact algorithm used was

*CONTACT_FORMING_ONE_WAY_SURFACE_TO_SURFACE_ID. The virtual

holder speed was set to 2000 mm/s, and the virtual stamping speed was set to 5000

mm/s. The strain distributions of the formed TRB specimens were obtained through

model. The accurate strain distributions could lay the foundation for the application of

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the forming limit margin field graph.

3.3. TRB bulging experiment

The TRB bulging experiment was performed on the setup shown in Fig. 7. The

TRB bulging experiments combined with the 3D DIC technology were carried out [23,

24]. The axisymmetric setup consisted of a pair of cameras, lights, a laser generator, a

die, a holder and a hemispherical punch. The inner diameter of the die and the blank

holder were both 105 mm, the punch diameter was 100 mm.

Fig. 7 TRB bulging experimental setup

The dimension of the TRB bulging specimen was 180 mm × 180 mm, as shown

in the Fig. 8. The specimen had constant thickness zone and thickness transition zone.

The constant thickness zone of TRB specimens contained the 1.2 mm zone and the 1.6

mm zone. The length of the 1.2 mm zone and 1.6 mm zone were both 70 mm. The

length of the thickness transition zone was 40 mm. Taking the centerline of the

specimen as the dividing line, the TRB bulging specimen was divided into thinner side

and thicker side along the rolling direction.

Fig. 8 Description of the TRB bulging specimen (Unit: mm)

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Prior to the experiments, the upper surfaces of the specimens were speckle

patterned by spraying in the form of a random pattern of black colored spots on a white

background. During the experiments, die moved downward to form the draw-bead on

the specimens with the stationary blank holder. The whole deformation processes of the

specimens were recorded by two cameras. The strain distributions of the specimens

were calculated by the digital image processing algorithm. In order to ensure pictures

could be recognized by the DIC system, the calibrated CCD cameras were placed in the

appropriate position above the sample surface. The cameras were triggered before the

specimens bulged. When the obvious necking or cracking occurred, the experiments

were terminated.

4. Results and Discussions

4.1. Simulation model verification

The cracking position and shape of the formed specimens were used as two simple

indexes to evaluate the accuracy of the model. The comparison between the simulation

result and the experimental result is illustrated in Fig. 9. The cracking position of the

TRB specimen was on the thinner side in the experiment in the simulation, the most

severe thinning area was found on the thinner side. As shown in the Fig. 9, the red area

(marked area in Fig. 9) was the most severely thinned area in the blank. In addition, the

deformation shape of the formed specimen was basically the same in the experimental

and simulation result. Therefore, it was initially indicated that the simulation model was

accurate.

Fig. 9 Deformation comparison of experimental and simulation result

To prove the accuracy of the final strain distributions, the strain paths at three

typical points were obtained by the simulation and the experiment, respectively. The

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first point was near the edge of cracking. The other two points and the first point were

on a line. The distance between the points was 8 mm. A comparison was made and the

results are shown in Fig. 10. The strain paths at the P1 and P2 showed that the deviation

between the major strains obtained by the simulation and those obtained through the

experiment was large when the minor strain was small. Then with the increase of the

minor strain, the deviation decreased gradually and coincided in a certain position. After

that, with the increase of the minor strain, the deviation increased. At the final minor

strain, the deviation between the final major strain obtained by the simulation and that

obtained through the experiment reached 9.97%, 5.86%, respectively. At the P3, the

deviation between the major strain obtained by the simulation and that obtained through

the experiment was 3.75% at the final minor strain. In the calculation of the forming

limit margin, the final strains were used. Therefore, it was concluded that final strains

obtained by simulation could be used.

Fig. 10 Strain paths comparison of experimental and simulation result: (a) schematic

diagram of cracking; (b) strain path at P1; (c) strain path at P2; (d) strain path at P3

In summary, the deformation of the specimens and the strain paths at the three

typical points in simulation were proved by the experiment. Thus, the final major and

minor strains could be used in the forming limit margin field graph to evaluate the

formability of the TRB.

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4.2. Quantitative evaluation of the formability before cracking

Three forming limit margin field graphs under three different punch travels of 31.9

mm, 33.0 mm and 34.1 mm were established. As shown in Fig. 11, the 3D FLDs are on

the left, and the forming margin field graphs are on the right.

In the 3D FLDs, as the punch travel increased, the strain state points were closer

to the FLS. All strain state points were below the FLS. In the forming margin field

graph, the forming margin distributions on the formed TRB part can be obtained

intuitively. According to the cloud ruler, the minimum forming margin was 0.030 when

the punch travel was 31.9 mm. When the punch travel was 33.0 mm, the minimum

forming margin was 0.017. When the punch travel reached 34.1 mm, the minimum

forming margin changed to 0.001. The minimum forming margins were positive, which

indicated that the formed TRB specimen was not cracked. As the punch travel increased,

the minimum forming margin decreased, indicating that the formability of the TRB

specimen decreased.

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Fig. 11 Forming limit margin field graphs under three different punch travels: (a)

punch travel of 31.9 mm; (b) punch travel of 33.0 mm; (c) punch travel of 34.1 mm

4.3. Cracking prediction and cracking degree evaluation

As shown in Fig. 12, the other three forming limit margin field graphs under three

different punch travels of 35.2 mm, 36.3 mm and 37.4 mm were established.

When the punch travel was 35.2 mm in the simulation, the forming limit margin

field graph can be seen from Fig. 12(a). In the 3D FLD, all strain state points were

below the FLS. Some of the strain state points near the FLS were displayed in red. The

thickness coordinates of these red strain state points were within the range of 1.2 mm

to 1.4 mm. In the forming margin field graph, the forming margin distributions on the

formed part can be obtained intuitively. According to the cloud ruler, the forming

margins of the red area were between 0.002 and 0.037, which indicated that the formed

specimen was not cracked but the ability of material in the red area to continue to

deform was insufficient. The red area was located on the thinner side.

When the punch travel reached 36.3 mm in the simulation, the forming limit

margin field graph was shown in Fig. 12(b). In the 3D FLD, some strain state points

marked red were located above the FLS. This indicated a cracking of the TRB specimen.

The thickness coordinates of these strain state points were between 1.2 mm and 1.4 mm.

In the forming margin field graph, negative values appeared in the forming margins of

all elements. The element with the minimum margin was on the thinner side. The

minimum margin value was -0.012, which meant that the formed specimen had cracked.

Some elements near the draw bead were marked red (see in Fig. 12(b)), but their margin

values were greater than zero, and there was no cracking.

When the punch continued to move, the travel reached 37.4 mm in the simulation.

17

The forming limit margin field graph was obtained, as shown in Fig.12(c). In the 3D

FLD, there were more strain state points marked red above the FLS. In the forming

margin field graph. The minimum margin value was -0.024. The absolute value of the

minimum margin became greater than its value at the 36.3 mm punch travel. The TRB

part cracked more seriously. The cracking position was on the thinner side. Similarly,

the elements marked red near the draw bead did not crack.

Fig. 12 Forming limit margin field graphs under three different punch travels: (a)

punch travel of 35.2 mm; (b) punch travel of 36.3 mm; (c) punch travel of 37.4 mm

At the three different punch travels, the minimum value of the forming margin, the

maximum value of the forming margin and the number of cracking elements are shown

in Table 3. As the punch travel increased, the minimum forming margin changed from

0.002, to -0.012, and to -0.024. The minimum forming margin value changed from

18

positive to negative, and the absolute value of the forming margin increased. The

number of cracking elements was increased from 0, to 8, and to 26. These results

indicated that with the increase of the punch travel, the cracking degree increased.

Table 3 Forming margins and the number of cracking elements of three punch travels

Travel (mm) Min margin Max margin Number of cracking

elements

35.2 0.002 0.247 0

36.3 -0.012 0.247 8

37.4 -0.024 0.247 26

The result of the forming limit margin field graph was verified experimentally. In

the experiments, the cracks occurred on the TRB parts when the punch travels were

within the range of 33 mm to 35 mm. The cracking positions were basically the same

on the thinner side. With the further increase of the punch travel, the cracking enlarged

further in the TRB bulging experiments. This phenomenon was consistent with the

findings of Zhang et al. [25]. The forming limit on the thinner side was smaller than

that on the thicker side. Under the same load, the stretching stress on the thinner side is

greater than that on the thicker side. As a result, the thickness thinning of the thinner

side was severe, and cracking eventually occurred.

5. Conclusions

In this paper, the influence of blank thickness on the TRB constitutive relation and

formability was considered. A calculation method of the forming margin was proposed

to quantify the formability. The forming limit margin field graph of TRB was

established. The major conclusions could be drawn as follows:

(1) The deformation and strain paths predicted by simulation agreed well with that

measured from experiments results, which indicated that the simulation model was

reliable. The constitutive relation of the TRB established by means of the

combination of the uniaxial tensile tests and the Lagrange polynomial interpolation

method was credible.

(2) The forming margin was a quantification of the formability. During the TRB

19

bulging process, the punch travel changed from 35.2mm, to 36.3mm, and to

37.4mm, the minimum forming margin value changed from 0.002, to -0.012, and

to -0.024. The TRB could continue to deform until TRB cracked, and eventually

the cracking became more serious. The cracking occurred on the thinner side.

(3) The simulation and experiments proved the forming limit margin field graph was

efficient. It was suitable for studying the formability of TRB and the prediction of

cracking in the forming processes. The 3D FLS obtained based on the forming limit

tests and polynomial fitting was reasonable.

Acknowledgements

This project is supported by National Natural Science Foundation of China (No.

51975202) and the Natural Science Foundation of Hunan Province (2019JJ30005).

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Figure captions

Fig. 1 A schematic of TRB technology (adapted from Ref. [4])

Fig. 2 3D FLS obtained by polynomial fitting

Fig. 3 3D FLD and definition of forming margin

Fig. 4 Flow chart of forming margin calculation for all elements

Fig. 5 True stress-true plastic strain curves of the TRB: (a) in the constant thickness

zone; (b) in the thickness transition zone

Fig. 6 Schematic of bulging modeling

Fig. 7 TRB bulging experimental setup

Fig. 8 Description of the TRB bulging specimen (Unit: mm)

Fig. 9 Deformation comparison of experimental and simulation result

Fig. 10 Strain paths comparison of experimental and simulation result: (a) schematic

diagram of cracking; (b) strain path at P1; (c) strain path at P2; (d) strain path at P3

Fig. 11 Forming limit margin field graphs under three different punch travels: (a) punch

travel of 31.9 mm; (b) punch travel of 33.0 mm; (c) punch travel of 34.1 mm

Fig. 12 Forming limit margin field graphs under three different punch travels: (a) punch

travel of 35.2 mm; (b) punch travel of 36.3 mm; (c) punch travel of 37.4 mm

24

Table captions

Table 1 Polynomial fitting coefficients of the 3D FLS

Table 2 Some mechanical properties of tested TRB

Table 3 Forming margins and the number of cracking elements of three punch travels

25

Declarations

Funding

This project is supported by National Natural Science Foundation of China (No.

51975202) and the Natural Science Foundation of Hunan Province (2019JJ30005).

Conflicts of interest/Competing interests

Conflict of Interest for all authors - None.

Availability of data and material

The raw/processed data and material required to reproduce these findings cannot

be shared at this time due to technical or time limitations.

Code availability

Not applicable.

Authors' contributions

Da Cai: Writing - Original Draft, Methodology. Hang Ou: Data Curation. Ming

Hu: Methodology, Software. Guangyao Li: Investigation. Junjia Cui: Writing -

Review & Editing, Funding acquisition.

Figures

Figure 1

A schematic of TRB technology (adapted from Ref. [4])

Figure 2

3D FLS obtained by polynomial �tting

Figure 3

3D FLD and de�nition of forming margin

Figure 4

Flow chart of forming margin calculation for all elements

Figure 5

True stress-true plastic strain curves of the TRB: (a) in the constant thickness zone; (b) in the thicknesstransition zone

Figure 6

Schematic of bulging modeling

Figure 7

TRB bulging experimental setup

Figure 8

Description of the TRB bulging specimen (Unit: mm)

Figure 9

Deformation comparison of experimental and simulation result

Figure 10

Strain paths comparison of experimental and simulation result: (a) schematic diagram of cracking; (b)strain path at P1; (c) strain path at P2; (d) strain path at P3

Figure 11

Forming limit margin �eld graphs under three different punch travels: (a) punch travel of 31.9 mm; (b)punch travel of 33.0 mm; (c) punch travel of 34.1 mm

Figure 12

Forming limit margin �eld graphs under three different punch travels: (a) punch travel of 35.2 mm; (b)punch travel of 36.3 mm; (c) punch travel of 37.4 mm